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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A276749 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(n^2) * x^k]^n / n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 3, 22, 749, 349707, 6584568222, 2542670826073083, 87482825374559636232439, 1084004198573118046271860417698544, 947790766920144318254338856937912501990845477, 546110521982991331256716555878135043551458467258822092049841, 1013482348116310649878997474896504367633097553028647215670516799670576593506574
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 749*x^4 + 349707*x^5 + 6584568222*x^6 + 2542670826073083*x^7 + 87482825374559636232439*x^8 +...
log(A(x)) = x + 5*x^2/2 + 58*x^3/3 + 2901*x^4/4 + 1744601*x^5/5 + 39505301300*x^6/6 + 17798649685552457*x^7/7 + 699862582655005078651885*x^8/8 + 9756037786370716942306622514588154*x^9/9 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^(n^2)*x^2 + 3^(n^2)*x^3 +...+ k^(n^2)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x)^2 + (x + 11*x^2 + 11*x^3 + x^4)^2/(1-x)^10/2 + (x + 502*x^2 + 14608*x^3 + 88234*x^4 + 156190*x^5 + 88234*x^6 + 14608*x^7 + 502*x^8 + x^9)^3/(1-x)^30/3 + (x + 65519*x^2 + 41932745*x^3 + 3572085255*x^4 + 85383238549*x^5 + 782115518299*x^6 + 3207483178157*x^7 + 6382798925475*x^8 + 6382798925475*x^9 + 3207483178157*x^10 + 782115518299*x^11 + 85383238549*x^12 + 3572085255*x^13 + 41932745*x^14 + 65519*x^15 + x^16)^4/(1-x)^68/4 + (x + 33554406*x^2 + 846416194536*x^3 + 1103881308184906*x^4 + 269025107855605626*x^5 + 21045399230106913746*x^6 + 695824003645512474376*x^7 + 11392907456028953400606*x^8 + 101955892318210543172751*x^9 + 531714261368950897339996*x^10 + 1685388700882132120106256*x^11 + 3334612565134607644610436*x^12 + 4179647109945703200884716*x^13 + 3334612565134607644610436*x^14 + 1685388700882132120106256*x^15 + 531714261368950897339996*x^16 + 101955892318210543172751*x^17 + 11392907456028953400606*x^18 + 695824003645512474376*x^19 + 21045399230106913746*x^20 + 269025107855605626*x^21 + 1103881308184906*x^22 + 846416194536*x^23 + 33554406*x^24 + x^25)^5/(1-x)^130/5 +...+ [Sum_{k=1..n^2} A008292(n^2,k) * x^k]^n / (1 - x^n)^(n^3+n) /n +...

Links

Crossrefs

Cf. A156170, A276748, A008292.

Programs

  • PARI
    {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(m^2) * x^k +x*O(x^n))^m / m ) ), n)}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m^2, A008292(m^2, k)*x^k/(1-x +Oxn)^(m^2+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n^2} A008292(n^2,k) * x^k / (1-x)^(n^2+1) ]^n / n ), where A008292 are the Eulerian numbers.

A277037 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n * 2^(n*k) * x^k]^n / n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 18, 484, 54630, 26638924, 53843811956, 442942117297000, 14725418961500037126, 1971239927985067569365772, 1060292226589575099894174194524, 2288290973515256950275126683431946552, 19795837218795604674370624304477542380054748, 685985356865646724678258830150265065104998427771576, 95174256167264272421248219248338459257647770713814222870312
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2016

Keywords

Comments

More generally, for fixed integer q, G(x,q) = exp( Sum_{n>=1} [Sum_{k>=1} k^n * q^(n*k) * x^k]^n / n ) is an integer series such that G(x,q) = exp( Sum_{n>=1} q^(n^2)/(1 - q^n*x)^(n^2+n) * [ Sum_{k=1..n} A008292(n,k) * q^(n*k-n) * x^k ]^n / n ), where A008292 are the Eulerian numbers.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 484*x^3 + 54630*x^4 + 26638924*x^5 + 53843811956*x^6 + 442942117297000*x^7 +...
such that the logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (2^n*x + 2^n*2^(2*n)*x^2 + 3^n*2^(3*n)*x^3 +...+ k^n*2^(k*n)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = 2*x/(1-2*x)^2 + 2^4*(x + 2^2*x^2)^2/(1-2^2*x)^6/2 + 2^9*(x + 4*2^3*x^2 + 2^6*x^3)^3/(1-2^3*x)^12/3 + 2^16*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)^4/(1-2^4*x)^20/4 + 2^25*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)^5/(1-2^5*x)^30/5 + 2^36*(x + 57*2^6*x^2 + 302*2^12*x^3 + 302*2^18*x^4 + 57*2^24*x^5 + 2^30*x^6)^6/(1-2^6*x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k ]^n / (1 - 2^n*x)^(n*(n+1))/n +...
Explicitly,
log(A(x)) = 2*x + 32*x^2/2 + 1352*x^3/3 + 214272*x^4/4 + 132616992*x^5/5 + 322738100480*x^6/6 + 3099838240135296*x^7/7 + 117796258487089512448*x^8/8 +...

Crossrefs

Cf. A277035, A156170, A277036, A008292.

Programs

  • PARI
    {a(n) = my(A=1,Oxn=x*O(x^n)); A = exp( sum(m=1,n+1, sum(k=1,n+1, k^m * 2^(m*k) * x^k +x*O(x^n) )^m / m )); polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m, A008292(m, k) * 2^(m*k) * x^k / (1 - 2^m*x +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k ]^n / (1 - 2^n*x)^(n*(n+1)) / n ), where A008292 are the Eulerian numbers.

A292501 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n * (1-x)^n / n ).

Original entry on oeis.org

1, 1, 3, 13, 91, 1119, 23235, 879361, 55447631, 6274018595, 1192773105789, 400761393446831, 231147252957096671, 231434829013884972151, 406000810484101907916927, 1216355994930424625967455929, 6474418584620388915674215696687, 58229572245447428847208518694227279, 936163501254507409972001699357677028097, 25330794407893091120626418701416294765820223, 1224635875718403110628189182372406488768960029317
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2017

Keywords

Comments

A060187(n,k) = Sum_{j=1..k} (-1)^(k-j) * binomial(n,k-j) * (2*j-1)^(n-1).
Note that exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ] / (1-x)^n * x^n/ n ) does not yield an integer series.
Conjecture: a(n)^(1/n^2) tends to 3^(1/4). - Vaclav Kotesovec, Oct 17 2020

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 91*x^4 + 1119*x^5 + 23235*x^6 + 879361*x^7 + 55447631*x^8 + 6274018595*x^9 + 1192773105789*x^10 + 400761393446831*x^11 + 231147252957096671*x^12 + 231434829013884972151*x^13 + 406000810484101907916927*x^14 + 1216355994930424625967455929*x^15 +...
RELATED SERIES.
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 305*x^4/4 + 5041*x^5/5 + 131477*x^6/6 + 5973311*x^7/7 + 436089793*x^8/8 + 55949083681*x^9/9 + 11863792842885*x^10/10 + 4395111080551775*x^11/11 + 2768928615166879025*x^12/12 + 3005637312940054635857*x^13/13 + 5680764740993004611483477*x^14/14 + 18239242940612856315412499071*x^15/15 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 +...+ (2*k-1)^n*x^k +...)^n * (1-x)^n/n,
or,
log(A(x)) = (x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 +...) * (1-x) +
(x + 3^2*x^2 + 5^2*x^3 + 7^2*x^4 + 9^2*x^5 +...)^2 * (1-x)^2/2 +
(x + 3^3*x^2 + 5^3*x^3 + 7^3*x^4 + 9^3*x^5 +...)^3 * (1-x)^3/3 +
(x + 3^4*x^2 + 5^4*x^3 + 7^4*x^4 + 9^4*x^5 +...)^4 * (1-x)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers of type B like so:
log(A(x)) = (x + x^2) / (1-x) +
(x + 6*x^2 + x^3)^2 / (1-x)^4/2 +
(x + 23*x^2 + 23*x^3 + x^4)^3 / (1-x)^9/3 +
(x + 76*x^2 + 230*x^3 + 76*x^4 + x^5)^4 / (1-x)^16/4 +
(x + 237*x^2 + 1682*x^3 + 1682*x^4 + 237*x^5 + x^6)^5 / (1-x)^25/5 +
(x + 722*x^2 + 10543*x^3 + 23548*x^4 + 10543*x^5 + 722*x^6 + x^7)^6 / (1-x)^36/6 +
(x + 2179*x^2 + 60657*x^3 + 259723*x^4 + 259723*x^5 + 60657*x^6 + 2179*x^7 + x^8)^7 / (1-x)^49/7 +...+
[ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n  / (1-x)^(n^2) * x^n/n +...

Links

Crossrefs

Cf. A292500, A156170, A276751, A276752, A156171, A155200, A060187.

Programs

  • PARI
    {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n, (2*k-1)^m * x^k +x*O(x^n))^m*(1-x)^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {A060187(n, k) = sum(j=1, k, (-1)^(k-j) * binomial(n, k-j) * (2*j-1)^(n-1))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = exp( sum(m=1,n+1, sum(k=0, m, A060187(m+1, k+1)*x^k)^m /(1-x +Oxn)^(m^2) * x^m/m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2) * x^n/n ), where A060187 are the Eulerian numbers of type B.

A370017 Expansion of g.f. exp( Sum_{n>=1} ( Sum_{k>=1} k^n*x^k )^n * (1-x)^n / n ).

Original entry on oeis.org

1, 1, 2, 6, 24, 134, 1054, 11848, 188498, 4229252, 132827660, 5831280558, 357547362450, 30623840955096, 3671208716930842, 616066177338250188, 145118327950242179484, 47979462271120402757058, 22322388348068543767280728, 14614554870662196578923073494, 13488493387943242211496467931272
Offset: 0

Views

Author

Paul D. Hanna, Feb 14 2024

Keywords

Comments

a(n)^(1/n^2) tends to 2^(1/4). - Vaclav Kotesovec, Feb 19 2024

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 134*x^5 + 1054*x^6 + 11848*x^7 + 188498*x^8 + 4229252*x^9 + 132827660*x^10 + 5831280558*x^11 + 357547362450*x^12 +...
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 531*x^5/5 + 5367*x^6/6 + 74313*x^7/7 + 1401295*x^8/8 + 36221143*x^9/9 + 1283583423*x^10/10 + ...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n * (1-x)^n/n,
or,
log(A(x)) = (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 +...)*(1-x) +
(x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + 5^2*x^5 +...)^2 * (1-x)^2/2 +
(x + 2^3*x^2 + 3^3*x^3 + 4^3*x^4 + 5^3*x^5 +...)^3 * (1-x)^3/3 +
(x + 2^4*x^2 + 3^4*x^3 + 4^4*x^4 + 5^4*x^5 +...)^4 * (1-x)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x) + (x + x^2)^2/(1-x)^4/2 + (x + 4*x^2 + x^3)^3/(1-x)^9/3 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^16/4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^25/5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^36/6 + ...
+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n^2)/n +...

Links

Crossrefs

Cf. A156170, A008292.

Programs

  • PARI
    {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m, A008292(m, k)*x^k +Oxn)^m / (1-x +Oxn)^(m^2) / m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) exp( Sum_{n>=1} ( Sum_{k>=1} k^n*x^k )^n * (1-x)^n / n ).
(2) exp( Sum_{n>=1} ( Sum_{k=1..n} A008292(n,k) * x^k )^n * (1-x)^(-n^2) / n ), where A008292 are the Eulerian numbers.
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